# Succinct approximate counting of skewed data

## Abstract

Practical data analysis relies on the ability to count observations of objects
succinctly and efficiently. Unfortunately the space usage of an exact estimator
grows with the size of the a priori set from which objects are drawn while the time
required to maintain such an estimator grows with the size of the data set. We
present static and on-line approximation schemes that avoid these limitations when
approximate frequency estimates are acceptable. Our Log-Frequency Sketch extends
the approximate counting algorithm of Morris [1978] to estimate frequencies with
bounded relative error via a single pass over a data set. It uses constant space
per object when the frequencies follow a power law and can be maintained in
constant time per observation. We give an (ε,δ)-approximation scheme which we
verify empirically on a large natural language data set where, for instance, 95
percent of frequencies are estimated with relative error less than 0.25 using fewer
than 11 bits per object in the static case and 15 bits per object on-line.